Given the universal set $U = \{1,2,3,4,5,6,7,8\}$, set $A = \{2,3,5,6\}$, set $\mathrm{B} = \{1,3,4,6,7\}$, then $\mathrm{A} \cap \mathrm{C}_{U}\mathrm{B} =$ (A) $\{2,5\}$ (B) $\{3,6\}$ (C) $\{2,5,6\}$ (D) $\{2,3,5,6,8\}$
Variables $x, y$ satisfy the constraints $\left\{\begin{array}{c}x + 2 \geq 0, \\ x - y + 3 \geq 0, \\ 2x + y - 3 \leq 0,\end{array}\right.$ then the maximum value of the objective function $Z = x + 6y$ is (A) 3 (B) 4 (C) 18 (D) 40
Let $x \in \mathbb{R}$. Then ``$|x - 2| < 1$'' is ``$x^2 + x - 2 > 0$'' a (A) sufficient but not necessary condition (B) necessary but not sufficient condition (C) necessary and sufficient condition (D) neither sufficient nor necessary condition
As shown in the figure, in circle O, M and N are trisection points of chord AB. Chords CD and CE pass through points M and N respectively. If $\mathrm{CM} = 2$, $\mathrm{MD} = 4$, $\mathrm{CN} = 3$, then the length of segment NE is (A) $\frac{8}{3}$ (B) 3 (C) $\frac{10}{3}$ (D) $\frac{5}{2}$
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, one of its asymptotes passes through the point $(2, \sqrt{3})$, and one focus of the hyperbola lies on the directrix of the parabola $y^2 = 4\sqrt{7}x$. Then the equation of the hyperbola is (A) $\frac{x^2}{21} - \frac{y^2}{28} = 1$ (B) $\frac{x^2}{28} - \frac{y^2}{21} = 1$ (C) $\frac{x^2}{3} - \frac{y^2}{4} = 1$ (D) $\frac{x^2}{4} - \frac{y^2}{3} = 1$
Given the function $\mathrm{f}(x) = 2^{|x-1|} - 1$ defined on $\mathbb{R}$ (where m is a real number) is an even function, let $\mathrm{a} = \mathrm{f}(\log_{0.5}3)$, $b = f(\log_2 5)$, $c = f(2m)$. Then the size relationship of $a, b, c$ is (A) $a < b < c$ (B) $a < c < b$ (C) $c < a < b$ (D) $c < b < a$
Given the function $F(x) = \left\{\begin{array}{l}2 - |x|, \quad x \leq 2 \\ (x - 2)^2, \quad x > 2\end{array}\right.$ and function $g(x) = b - f(2 - x)$, where $b \in \mathbb{R}$. If the function $y = f(x) - g(x)$ has exactly 4 zeros, then the range of $b$ is (A) $\left(\frac{7}{4}, +\infty\right)$ (B) $\left(-\infty, \frac{7}{4}\right)$ (C) $\left(0, \frac{7}{4}\right)$ (D) $\left(\frac{7}{4}, 2\right)$
In $\triangle \mathrm{ABC}$, the angles $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and their opposite sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ respectively. Given that the area of $\triangle \mathrm{ABC}$ is $3\sqrt{15}$, $b - c = 2$, $\cos A = -\frac{1}{4}$, then the value of $a$ is .
In isosceles trapezoid ABCD, $\mathrm{AB} \parallel \mathrm{DC}$, $\mathrm{AB} = 2$, $\mathrm{BC} = 1$, $\angle \mathrm{ABC} = 60°$. Moving points E and F are on segments BC and DC respectively, with $\overrightarrow{\mathrm{BE}} = \lambda\overrightarrow{\mathrm{BC}}$, $\overrightarrow{\mathrm{DF}} = \frac{1}{9\lambda}\overrightarrow{\mathrm{DC}}$. Then the minimum value of $\overrightarrow{\mathrm{AE}} \cdot \overrightarrow{\mathrm{AF}}$ is .
Given the function $f(x) = \sin^2 x - \sin^2\left(x - \frac{\pi}{6}\right)$, $x \in \mathbb{R}$. (I) Find the minimum positive period of $f(x)$; (II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$.
To promote the development of table tennis, a certain table tennis competition allows athletes from different associations to form teams. There are 3 athletes from Association A, of which 2 are seeded players, and 5 athletes from Association B, of which 3 are seeded players. Randomly select 4 people from these 8 athletes to participate in the competition. (I) Let A be the event ``exactly 2 seeded players are selected, and these 2 seeded players are from the same association''. Find the probability of this event. (II) Let X be the number of seeded players among the 4 selected people. Find the probability distribution and mathematical expectation of the random variable X.
As shown in the figure, in the quadrangular prism $\mathrm{ABCD} - A_1B_1C_1D_1$, the lateral edge $AA_1 \perp$ base $\mathrm{ABCD}$, $\mathrm{AB} \perp \mathrm{AC}$, $\mathrm{AB} = 1$, $\mathrm{AC} = AA_1 = 2$, $AD = CD = \sqrt{5}$, and points M and N are the midpoints of $B_1C$ and $D_1D$ respectively. (I) Prove: $\mathrm{MN} \parallel$ plane ABCD (II) Find the sine value of the dihedral angle $D_1 - AC - B_1$; (III) Let E be a point on edge $A_1B_1$. If the sine value of the angle between line NE and plane ABCD is $\frac{1}{3}$, find the length of segment $A_1E$.
Given the sequence $\{a_n\}$ satisfies $a_{n+2} = qa_n$ (where q is a real number and $q \neq 1$), $n \in \mathbb{N}^*$, $a_1 = 1$, $a_2 = 2$, and $a_2 + a_3$, $a_3 + a_4$, $a_4 + a_5$ form an arithmetic sequence. (I) Find the value of q and the general term formula of $\{a_n\}$; (II) Let $b_n = \frac{\log_2 a_{2n}}{a_{2n-1}}$, $n \in \mathbb{N}^*$. Find the sum of the first n terms of the sequence $\{b_n\}$.
Given an ellipse with left focus $\mathrm{F}(-c, 0)$ and eccentricity $\frac{\sqrt{3}}{3}$. Point M is on the ellipse and in the first quadrant. The line segment of line FM intercepted by the circle $x^2 + y^2 = \frac{b^2}{4}$ has length c, and $|FM| = \frac{4\sqrt{3}}{3}$. (I) Find the slope of line FM; (II) Find the equation of the ellipse; (III) Let P be a moving point on the ellipse. If the slope of line FP is greater than $\sqrt{2}$, find the range of the slope of line OP